On a two-phase Serrin-type problem and its numerical computation

Abstract

We consider an overdetermined problem of Serrin-type with respect to an operator in divergence form with piecewise constant coefficients. We give sufficient condition for unique solvability near radially symmetric configurations by means of a perturbation argument relying on shape derivatives and the implicit function theorem. This problem is also treated numerically, by means of a steepest descent algorithm based on a Kohn-Vogelius functional.

Symmetry Breaking Solutions for a Two-Phase Overdetermined Problem of Serrin-Type

“…Proof. Since the eigenvalues of (4.2) have been computed in [CY1], in what follows we only need to check that the right-hand side in (4.2) vanishes if and only if k = 0. Furthermore, since N (Y 0,i ) = 0 by construction, it will suffice to show that N (Y k,i ) = 0 for k ∈ N. To this end, let ϕ denote the numerator in the right-hand side of (4.2), that is…”

“…By the method of separation of variables, it can be shown that the spherical harmonics form an orthonormal basis of eigenfunctions of N in L 2 (∂Ω 2 ). The eigenvalues of N have been computed in [CY1].…”